Sunday, October 31, 2010

The mathematician says, “I believe it’s 4, but I’ll have to prove it.”

The statistician says, “The sample is too small to give a precise answer, but based on the data set, there is a high probability it is somewhere between 3 and 5.”

The engineer says, “The answer is 4, but I’ll have to add a safety factor so we’ll call it 5.”

Sales and Marketing Department

A group of Sales and Marketing personnel are charged with measuring the height of a flagpole outside their office. They soon head outside and start climbing all over each other in their suits and ties trying to get the tape measure to the top of the pole.

Seeing the difficulty the group is having, an engineer that is passing by offers to help. He quickly assesses the situation, lifts up the pole, and removes it from the base. Then he lays it down on the grass and measures it end to end. When he’s done, he puts it back up, gives his measurement to the group and walks away.

After the engineer is gone, one of the marketing guys turns to the group and says, “That’s just like an engineer… We asked for the height and he gave us the length.”

The Lawyer Genie

An engineer bought a shiny brass lamp at a garage sale. After he paid for it, the old woman running the sale took him aside and give him a warning: “This is a genuine genie lamp. Rub it and you will get 3 wishes, but beware… this lamp was designed by a lawyer so whatever you wish for, every lawyer in the world will receive twice over.”

Curious, the engineer took the lamp home and proceeded to make his first wish. “I wish for a Porsche 911.” He went to the window and looked outside. Sure enough, there was a brand new Porsche in his driveway. Then he looked down the street at a lawyer’s house and noticed that there were two brand new Porsches in his driveway.

Not overly concerned, he went and rubbed the lamp again. He said, “I wish for 10 million dollars.” Still confident from his first wish, he was sure he now had $10 million in his bank account and every lawyer in the world now had $20 million.

Not wanting to waste any time, the engineer immediately rubbed the lamp again. For his last wish he said, “I wish I could donate a kidney.”

Saturday, October 30, 2010

I'd love to attend, but there's yardwork in my plans today, thanks to the leafstorm that ripped through America's Northeast this week. It was the most beautiful, colorful storm I ever encountered, sunny with a strong chance of killing my grass, so first things first. DAMN YOU Oaks and Maples! Otherwise, I'd have loved to have attended this rally that goes down today:

Washington (CNN) -- Comedy Central hosts Jon Stewart and Stephen Colbert will hold the "Rally to Restore Sanity" and the "March to Keep Fear Alive" on the National Mall on Saturday, with thousands expected to flood in from across the country.

The event comes months after conservative Fox News host Glenn Beck held a "Rally to Restore Honor" in Washington, in which thousands of conservatives and Tea Party faithful attended.

While Stewart has insisted that the rally is not a result of Beck's march, the event's Web pages appear to copy the look of Beck's event, and many say it's a direct response to Beck's often-controversial comments.

The rallies are, in essence, an extension of both comedians' shows. The "Daily Show" and "The Colbert Report" routinely blast Republicans, Democrats and the media for hyperpartisan attacks and outrageous and inflammatory innuendo.

"Republicans love America. They just seem to hate about 50 percent of the people who live in it," Stewart said on CNN's "Larry King Live" last week. "Democrats, for their thing, it's always: They love this country, they just somehow wish it were a different country. With Democrats, it's like, 'America is the greatest country in the world. [But] have you seen Finland's health care system?'"

Stewart began the week in Washington, hosting his show from a downtown studio. He interviewed President Obama on Wednesday, grilling him on the upcoming election and why Democrats seem to be disenchanted with the administration.

Colbert recently blasted Sean Bielat, the Republican running against Democratic Rep. Barney Frank, and Ken Buck, the Republican running in Colorado's Senate race, for their views on gays and lesbians. Buck linked being gay to alcoholism while Bielat said that gays and lesbians shouldn't serve in the military, just as short people cannot.

In his typical deadpan shtick, Colbert said, "Sean Bielat and Ken Buck can't help being politically opportunistic election year gay-baiters. They were born that way."

Humor aside, Stewart said the rally is for people who are "tired of their reflection in the media as being a divided country and a country that's ideological and conflicted and fighting, this is for those people."

"It's going to shock, maybe not even just this world, other worlds," he joked. "Maybe aliens."

The rally's popularity has gone viral -- from Facebook pages to local rallies in cities across the country. Several groups, including the liberal media blog Huffington Post and even daytime talk show queen Oprah Winfrey, have offered to transport people to Washington for the event.

One of the many in the crowd will be Heidi Thomas, a soccer mom from Virginia who is gathering some friends and heading to the rally.

"As soon as I heard about this, I knew I was going," she told CNN. "It was immediate. ... I got to the point where I was feeling very grumpy every night I'd go to bed because of the political tone and nature of the country right now."

Thomas said that after watching Stewart's show, she'd go to bed in a better mood, "because I'd feel like I wasn't going insane," adding that there were "actually sane people out there."

But things could turn to insanity when it comes to the bathroom situation.

Rally organizers are having trouble finding portable toilets for those attending, WTOP.com reports. Their event occurs the day before the Marine Corps Marathon, and organizers of that event have locked the 800 they have rented until Sunday morning.

Thomas is hopeful the rally will expose a segment of the population fed up with the political fringes.

"I think it's important to show that there is a majority or a lot of people who are feeling more moderate," she said. "It's just frustrating to me.

While Stewart says the rally won't be a partisan event, one media analyst wonders whether the rally will change Stewart from satirist to political activist.

"To me, the bottom-line question is this: Is this going to be a fun Saturday event with a lot of laughs?" said Howard Kurtz, host of CNN's Sunday program "Reliable Sources" and Washington bureau chief for The Daily Beast. "Or is it going to be something that, while wrapped in humor, is going to make a serious political point about folks in the middle -- moderates who are alienated ... by the partisan shouting on both sides?

"If that happens, Jon Stewart will have made a serious point while still having good fodder for 'The Daily Show.'"

UPDATE (Nov. 3, 2010): Alan Boyle at CosmicLog has a nice piece up here showing a satellite view of the event:

"Comedy Central's park permit anticipated that 60,000 spectators would show up, while Colbert estimated the attendance at 6 billion. This satellite picture of the mall, captured by the GeoEye-1 satellite during the rally, gives you an astronaut's-eye view of the crowd from 423 miles up. Maybe those are 6 billion ants ..."

Where n is the number of variables and L is the number of bits of input to the algorithm, Karmarkar's algorithm requires O(n3.5L) operations on O(L) digit numbers, as compared to O(n6L) such operations for the ellipsoid algorithm. The runtime of Karmarkar's algorithm is thus O(n3.5L2lnLlnlnL) using FFT-based multiplication (see Big O notation).

Karmarkar's algorithm falls within the class of interior point methods: the current guess for the solution does not follow the boundary of the feasible set as in the simplex method, but it moves through the interior of the feasible region and reaches the optimal solution only asymptotically.

Contents

The Algorithm

The algorithm determines the next feasible direction toward optimality and scales back by a factor 0 < γ ≤ 1.

Karmarkar's algorithm is rather complicated. A simplified version, called the affine-scaling method, proposed and analyzed by others, can be described succinctly as follows. Note that the affine-scaling algorithm, while efficient in practice, is not a polynomial time algorithm.

Algorithm Affine-ScalingInput: A, b, c, x0, stopping criterion, γ.

do whilestopping criterionnot satisfiedifthenreturn unboundedend if

end do

"←" is a loose shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.

"return" terminates the algorithm and outputs the value that follows.

Example

Example solution

Consider the linear program

maximize

x1

+

x2

subject to

2px1

+

x2

p2 + 1,

That is, there are 2 variables x1,x2 and 11 constraints associated with varying values of p. This figure shows each iteration of the algorithm as red circle points. The constraints are shown as blue lines.

Patent controversy

At the time he discovered the algorithm, Narendra Karmarkar was employed by AT&T and they realized that his discovery could be of practical importance. In April 1985, AT&T promptly applied for a patent on Karmarkar's algorithm and that became more fuel for the ongoing controversy over the issue of software patents.[1] This left many mathematicians uneasy, such as Ronald Rivest (himself one of the holders of the patent on the RSA algorithm), who expressed the opinion that research proceeded on the basis that algorithms should be free. Even before the patent was actually granted, it seemed that there was prior art that might have been applicable.[2] Mathematicians who specialize in numerical analysis such as Philip Gill and others showed that Karmarkar's algorithm is actually equivalent to a projected Newton barrier method with a logarithmic barrier function, if the parameters are chosen suitably.[3] However, some[who?] say Gill's argument is flawed, insofar as the method they describe does not even qualify as an "algorithm", since it requires choices of parameters that don't follow from the internal logic of the method, but rely on external guidance, essentially from Karmarkar's algorithm.[citation needed] Methods referred to by Gill were widely used for nonlinear programming since the 1960s. In fact, one well-known book first published in 1968 described the technique specifically in the context of linear programming.[4] Nevertheless, the patent was eventually granted as U.S. Patent 4,744,026: "Methods and apparatus for efficient resource allocation" in May 1988. The patent, however, proved to be of limited commercial value to AT&T. They built up the KORBX system, an 8-processor Alliant computer incorporating linear programming software using Karmarkar's algorithm, priced at US$8.9 million each, and unsurprisingly they only managed to sell two such systems.[2] Opponents of software patents have further alleged that the patents ruined the positive interaction cycles that previously characterized the relationship between researchers in linear programming and industry, and specifically it isolated Karmarkar himself from the network of mathematical researchers in his field. [5]

The patent itself expired in April 2006, and the algorithm is presently in the public domain.

Tuesday, October 26, 2010

"The conviction that a TOE can be constructed might seem wishful thinking, but there are reasons to believe that a divergence-free TOE exists but requires a profound rethinking of the fundamental concepts of physics. A potential solution could be realized by finding a way to transform the Einstein-Hilbert action into a holographic description, and to apply the Feynman sum approach not to the Einstein-Hilbert action, but to the corresponding holographic action. The hope is that the vastly reduced number of paths in the holographic Feynman sum would tame the divergences."

"Be assured that we have no clue of what lies within a quark. Quarks might very well be point-like, and as such they are described in the standard model. But if they had an internal structure of dimensions testable with the energetic probes we are using (other quarks or gluons), we would be experiencing yet again the marvel of Lord Rutherford, as he saw the alpha particles with which he was bombarding a gold foil rebounding back as if they had hit something really hard inside the atom."

Monday, October 25, 2010

Researchers in Fairbanks Alaska announced last week that they have discovered a superconductor which will operate at room temperature.

A Princeton plasma physicist is at the beach when he discovers an ancient looking oil lantern sticking out of the sand. He rubs the sand off with a towel and a genie pops out. The genie offers to grant him one wish. The physicist retrieves a map of the world from his car an circles the Middle East and tells the genie, 'I wish you to bring peace in this region'.

After 10 long minutes of deliberation, the genie replies, 'Gee, there are lots of problems there with Lebanon, Iraq, Israel, and all those other places. This is awfully embarrassing. I've never had to do this before, but I'm just going to have to ask you for another wish. This one is just too much for me'.

Taken aback, the physicist thinks a bit and asks, 'I wish that the Princeton tokamak would achieve scientific fusion energy break-even.'

After another deliberation the genie asks, 'Could I see that map again?'

Two atoms bump into each other. One says 'I think I lost an electron!' The other asks, 'Are you sure?', to which the first replies, 'I'm positive.'

A physics student was hit by a brick falling from a house. He fainted, but came to after a while and started smiling. The onlookers were worried, so they asked him why the smile. "I just realized how lucky I am because the kinetic energy is only half m v squared."

There is this farmer who is having problems with his chickens. All of the sudden, they are all getting very sick and he doesn't know what is wrong with them. After trying all conventional means, he calls a biologist, a chemist, and a physicist to see if they can figure out what is wrong. So the biologist looks at the chickens, examines them a bit, and says he has no clue what could be wrong with them. Then the chemist takes some tests and makes some measurements, but he can't come to any conclusions either. So the physicist tries. He stands there and looks at the chickens for a long time without touching them or anything. Then all of the sudden he starts scribbling away in a notebook. Finally, after several gruesome calculations, he exclaims, 'I've got it! But it only works for spherical chickens in a vacuum.'

A young physicist, upon learning that he was denied tenure after six productive years at a University in San Francisco, requested a meeting with the Provost for an explanation, and a possible appeal.

At the meeting, the Provost told the young physicist, "I'm sorry to tell you that the needs of the University have shifted somewhat, during the past six-years leading up to your tenure decision. In point of fact, what we now require is a female, condensed-matter experimentalist. Unfortunately, you are a male, high-energy theorist!"

Dejected but not defeated, the young physicist thought for a moment about the implications of the Provost's words. "Sir," he said, "I would be willing to convert in two of the three categories you mention, but ... I'll never agree to become an experimentalist!"

Friday, October 22, 2010

A mathematician and an engineer are sitting at a table drinking when a very beautiful woman walks in and sits down at the bar.

The mathematician sighs. "I'd like to talk to her, but first I have to cover half the distance between where we are and where she is, then half of the distance that remains, then half of that distance, and so on. The series is infinite. There'll always be some finite distance between us."

The engineer gets up and starts walking. "Ah, well, I figure I can get close enough for all practical purposes."

************

A mathematician is a device for turning coffee into theorems
... Alfréd Rényi

************

A topologist is a mathematician who can't tell the difference between a doughnut and a coffee mug.

************

Did you know that all numbers are interesting? What’s that? You don’t believe me? Well I have a proof. Suppose not every number is interesting. Then let n be the smallest uninteresting number. That’s a rather interesting property isn’t it?
... Ron Graham

Q: What is the difference between a mathematician and a philosopher?
A: The mathematician only needs paper, pencil, and a trash bin for his work - the philosopher can do without the trash bin...

Q: What is the difference between a Ph.D. in mathematics and a large pizza?
A: A large pizza can feed a family of four.

When the math professor's wife returns home from work, she finds an envelope on the living room table. She opens it and finds a letter from her husband:

My dearest wife,

We have been married for nearly thirty years, and I still love you as much as on the day I proposed. You must realize, however, that you are now 54 years old and no longer able to satisfy certain needs I still have. I very much hope that you are not hurt to learn that, while you're reading this, I'm in a hotel room with an 18-year-old freshman girl from my calculus class. I'll be home before midnight.

Your husband, who will never stop loving you.

When the professor returns from the hotel shortly before midnight, he also finds an envelope in the living room. He opens it and reads:

My beloved husband,

You may recall that you, too, are 54 years old and no longer able to satisfy certain needs I still have. I thus hope that you are not hurt to learn that, while you're reading this, I am in a hotel room with the 18-year-old pool boy.

Your loving wife.

P.S. As a mathematician, you are certainly aware of the fact that 18 goes into 54 many more times than 54 goes into 18. Therefore, don't stay up and wait for me.

The integral of e raised to the power of x equals the function of u raised to the power of n.

(Write it out in notation to see the joke)

Did you really write it out? You didn't do that in your head? ;-)

True story:
A student walked into his discrete math class late and in order not to interrupt he put his late slip on the teacher's desk furtively without the teacher noticing. The teacher noticed the slip on his desk afterwards. He commented "I see you put this slip on my desk without me noticing. I guess that's why they call this class discrete mathematics."

There is a shipwreck, and the only three survivors are a Doctor, a Lawyer, and a Mathematician, in a rowboat.

After some time drifting about the seas, eventually they get get to talking and get to know each other. One day the doctor asks, "Is it better to have a wife or a girlfriend? I would say it's better to have a wife. I work long hard and emotional hours, and it's really great to have a caring wife who cooks great meals, cleans my clothes, and expertly manages our home and children."

The lawyer says, "I think it's better to have a girlfriend. I'm a Divorce Lawyer and the cost to the man in Divorce is so extreme I don't see where having a wife is worth the risk."

The mathematician says, "I think it's better to have both."

"What !?" say the doctor and lawyer. "Why?"

"Because," the mathematician says, "You can tell your wife you're working late, and your girlfriend you need to spend time with your family, which gives you more time to work on proving the Riemann Hypothesis !"

Wednesday, October 20, 2010

Who was the smartest of the smarties? Gauss? Young? Einstein? Dirac? Witten? Smolin? Of ALL time? How about Nick Bourbaki? It's been said he had the power of 15 fine minds.

If he were alive today, would Peter Woit let him post on Not Even Wrong, or moderate him heavily? A mystery!

From Wikipedia:

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still discussed.

While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki (Association of Collaborators of Nicolas Bourbaki), which has an office at the École Normale Supérieure in Paris.

Books by Bourbaki

Aiming at a completely self-contained treatment of the core areas of modern mathematics based on set theory, the group produced the Elements of Mathematics (Éléments de mathématique) series, which contains the following volumes (with the original French titles in parentheses):

The book Variétés différentielles et analytiques was a fascicule de résultats, that is, a summary of results, on the theory of manifolds, rather than a worked-out exposition. A final volume IX on spectral theory (Théories spectrales) from 1983 marked the presumed end of the publishing project; but a further commutative algebra fascicle was produced in 1998.

While several of Bourbaki's books have become standard references in their fields, some have felt that the austere presentation makes them unsuitable as textbooks.[1] The books' influence may have been at its strongest when few other graduate-level texts in current pure mathematics were available, between 1950 and 1960.[2]

It is frequently claimed[by whom?] that the use of the blackboard bold letters for the various sets of numbers was first introduced by the group. There are several reasons to doubt this claim.[3]

Influence on mathematics in general

The emphasis on rigour may be seen as a reaction to the work of Henri Poincaré,[4] who stressed the importance of free-flowing mathematical intuition, at a cost of completeness in presentation. The impact of Bourbaki's work initially was great on many active research mathematicians world-wide.

It provoked some hostility, too, mostly on the side of classical analysts; they approved of rigour but not of high abstraction. Around 1950, also, some parts of geometry were still not fully axiomatic — in less prominent developments, one way or another, these were brought into line with the new foundational standards, or quietly dropped. This undoubtedly led to a gulf with the way theoretical physics is practiced.[5]

Bourbaki's direct influence has decreased over time.[5] This is partly because certain concepts which are now important, such as the machinery of category theory, are not covered in the treatise. The completely uniform and essentially linear referential structure of the books became difficult to apply to areas closer to current research than the already mature ones treated in the published books, and thus publishing activity diminished significantly from the 1970s.[6] It also mattered that, while especially algebraic structures can be naturally defined in Bourbaki's terms, there are areas where the Bourbaki approach was less straightforward to apply.[citation needed]

On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed.[7] This is particularly true for the less applied parts of mathematics.

The Bourbaki seminar series founded in post-WWII Paris continues. It is an important source of survey articles, written in a prescribed, careful style. The idea is that the presentation should be on the level of absolute specialists, but for an audience which is not specialized in the particular field.

The original goal of the group had been to compile an improved mathematical analysis text; it was soon decided that a more comprehensive treatment of all of mathematics was necessary. There was no official status of membership, and at the time the group was quite secretive and also fond of supplying disinformation. Regular meetings were scheduled, during which the whole group would discuss vigorously every proposed line of every book. Members had to resign by age 50.[9]

The atmosphere in the group can be illustrated by an anecdote told by Laurent Schwartz. Dieudonné regularly and spectacularly threatened to resign unless topics were treated in their logical order, and after a while others played on this for a joke. Godement's wife wanted to see Dieudonné announcing his resignation, and so on one occasion while she was there Schwartz deliberately brought up again the question of permuting the order in which measure theory and topological vector spaces were to be handled, to precipitate a guaranteed crisis.

The name "Bourbaki" refers to a French general Charles Denis Bourbaki;[10] it was adopted by the group as a reference to a student anecdote about a hoax mathematical lecture, and also possibly to a statue. It was certainly a reference to Greek mathematics, Bourbaki being of Greek extraction. It is a valid reading to take the name as implying a transplantation of the tradition of Euclid to a France of the 1930s, with soured expectations.[11]

Appraisal of the Bourbaki perspective

The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Göttingen school, particularly Hilbert and the modern algebra school of Emmy Noether, Artin and van der Waerden. It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics. But always through a transforming process of reception and selection — their ability to sustain this collective, critical approach has been described as "something unusual".[12]

The following is a list of some of the criticisms commonly made of the Bourbaki approach:[13]

algorithmic content is not considered on-topic and is almost completely omitted[14]

Measure theory is developed from a functional analytic perspective. Taking the case of locally compact measure spaces as fundamental focuses the presentation on Radon measures and leads to an approach to measurable functions that is cumbersome, especially from the viewpoint of probability theory.[17] However, the last chapter of the book addresses limitations, especially for use in probability theory, of the restriction to locally compact spaces.

Dieudonné as speaker for Bourbaki

Public discussion of, and justification for, Bourbaki's thoughts has in general been through Jean Dieudonné (who initially was the 'scribe' of the group) writing under his own name. In a survey of le choix bourbachique written in 1977, he did not shy away from a hierarchical development of the 'important' mathematics of the time.

He also wrote extensively under his own name: nine volumes on analysis, perhaps in belated fulfillment of the original project or pretext; and also on other topics mostly connected with algebraic geometry. While Dieudonné could reasonably speak on Bourbaki's encyclopedic tendency, and tradition (after innumerable frank tais-toi, Dieudonné! ("Hush, Dieudonné!") remarks at the meetings), it may be doubted whether all others agreed with him about mathematical writing and research. In particular Serre has often championed greater attention to problem-solving, within number theory especially, not an area treated in the main Bourbaki texts.

Dieudonné stated the view that most workers in mathematics were doing ground-clearing work, in order that a future Riemann could find the way ahead intuitively open. He pointed to the way the axiomatic method can be used as a tool for problem-solving, for example by Alexander Grothendieck. Others found him too close to Grothendieck to be an unbiased observer. Comments in Pal Turán's 1970 speech on the award of a Fields Medal to Alan Baker about theory-building and problem-solving were a reply from the traditionalist camp at the next opportunity[21][not in citation given], Grothendieck having received the previous Fields Medal in absentia in 1966.

Bourbaki's influence on mathematics education

In the longer term, the manifesto of Bourbaki has had a definite and deep influence. In secondary education the new math movement corresponded to teachers influenced by Bourbaki. In France the change was secured by the Lichnerowicz Commission.[22]

The influence on graduate education in pure mathematics is perhaps most noticeable in the treatment now current of Lie groups and Lie algebras. Dieudonné at one point said 'one can do nothing serious without them', for which he was reproached; but the change in Lie theory to its everyday usage owes much to the type of exposition Bourbaki championed. Beforehand Jacques Hadamard despaired of ever getting a clear idea of it.

See also

Notes

^Confronted by the task of appraising a book by N. Bourbaki, this reviewer feels as if he were required to climb the Nordwand of the Eiger. The presentation is austere and monolithic. The route is beset by scores of definitions, many of them apparently unmotivated. Always there are hordes of exercises to be worked through painfully. One must be prepared to make constant cross-references to the author's many other works.Hewitt, Edwin (1956). "Review: Espaces vectoriels topologiques". Bulletin of the American Mathematical Society62: 507–508. doi:10.1090/S0002-9904-1956-10042-6.[1]

^...by 1958 when the original six books were completed, the first few of these books were already almost 20 years out of date.[2]

^ (1) the symbols do not appear in Bourbaki publications (rather, ordinary bold is used) at or near the era when they began to be used elsewhere, for instance, in typewritten lecture notes from Princeton University (achieved in some cases by overstriking R or C with I), and (an apparent first) typeset in Gunning and Rossi's textbook on several complex variables;[citation needed] (2) Jean-Pierre Serre, a member of the Bourbaki group, has publicly inveighed against the use of "blackboard bold" anywhere other than on a blackboard.[citation needed]

^Bourbaki came to terms with Poincaré only after a long struggle. When I joined the group in the fifties it was not the fashion to value Poincaré at all. He was old-fashioned. Pierre Cartier interviewed by Marjorie Senechall. "The Continuing Silence of Bourbaki". Mathematical Intelligencer19: 22–28. 1998.[3]

^ The minutes are in the Bourbaki archives — for a full description of the initial meeting consult Liliane Beaulieu in the Mathematical Intelligencer.

^ This resulted in a complete change of personnel by 1958; see Robert Mainard paper cited below. However, the Aubin paper cited below quotes the historian Liliane Beaulieu as never having found written affirmation of this rule.

^ It is said that Weil's wife Evelyne supplied Nicolas. (Mentioned by McCleary (PDF). This is more or less confirmed by Robert Mainard((PDF), a long article in French, which gives numerous further details: why N?, and the prank lecture of Raoul Husson in a false beard that gave rise to Bourbaki's theorem). They married in 1937, she having previously been with de Possel; who then unsurprisingly left the group.

^Pierre Cartier, a Bourbaki member 1955–1983, comments explicitly on several of these points (The Continuing Silence of Bourbaki, article from the Mathematical Intelligencer): ...essentially no analysis beyond the foundations: nothing about partial differential equations, nothing about probability. There is also nothing about combinatorics, nothing about algebraic topology, nothing about concrete geometry. And Bourbaki never seriously considered logic. Dieudonné himself was very vocal against logic. Anything connected with mathematical physics is totally absent from Bourbaki's text.

^Tim Gowers discusses at length the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories in his The Two Cultures of Mathematics (PDF).

^Lennart Carleson spoke of this in an interview (Infomat August 2006 (PDF)): ...that book [from 1968] was written mostly as a way to encourage the teachers to stay with established values. That was during the Bourbaki and New Math period and mathematics was really going to pieces, I think. The teachers were very worried and they had very little backing.

^ Heinz König: The traditional abstract measure theory which emerged from the achievements of Borel and Lebesgue in the first two decades of the 20th century is burdened with its total limitation to sequential procedures and its neglect of regularity. The alternative theory due to Bourbaki which arose in the middle of the century was able to relieve these burdens, but produced new ones. In particular its fundamental turn to inner regularity, based on the profound role of compactness, was done with the inappropriate weapons from the outer arsenal, which subsequently enforced that unfortunate construction named the essential one. All this produced serious obstacles against a unified theory of measure and integration, for example for the notion of signed measures, the formation of products and for the representation theorems of Daniell-Stone and Riesz types.[5]

^ Pierre Cartier, in the article cited above, is quoted as later saying The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith.

^ In the French context it has been said that geometry was in effect exiled from secondary teaching: Pour ce qui est des années 1960, l’effet de la réforme dite des mathématiques modernes sur l’enseignement de la géométrie est bien connu : si Dieudonné, comme Bourlet finalement, lance "A bas Euclide", le résultat n’est pas l’élaboration d’une géométrie plus expérimentale, plus intuitive. C’est l’effacement de la géométrie derrière l’algèbre linéaire et la quasi-disparition de l’enseignement de la géométrie élémentaire au collège et au lycée pour une dizaine d’années.—"As for the 1960s, the effect of this reform of modern mathematics on the teaching of geometry is well-known: if Dieudonné, like Bourlet finally, says "push Euclid back," the result is not the development of a geometry that is more experimental, more intuitive. It's the erasure of geometry behind linear algebra, and the quasi-disappearance of the teaching of elementary geometry in high school, for ten years."[6]

About Me

My weblog is named "Multiplication by Infinity", because "Division by Zero" was taken ... and "Division by Infinity" makes me feel very small ... Steven Colyer's Musings in Mathematical Physics and its Effects on Humanity and other Lifeforms.... And Pure Mathematics, Computer Science, Applied Mathematics, Experimental Physics, Engineering, Astronomy (not Cosmology so much), Space Exploration and Lunar Colonization.
I am a Rutgers 1979 Mechanical Engineer (Pi Tau Sigma) and Rutgers 1989 MBA.
("I study Politics and War that my children may study Mathematics and Philosophy."
- 2nd U.S. President John Adams)
I've already studied enough Politics and War and Economics for one lifetime, and so it's time for Math and Science